The_Cambridge_Handbook_of_Physics_Formulas

64 Dynamics and mechanics
3.2 Frames of reference
Galilean transformations
Time and
position a
r = r ′ +vt (3.1)
t = t ′ (3.2)
r,r ′
position in frames S
and S ′
v velocity of S ′ in S
t,t ′ time in S and S ′
Velocity u = u ′ +v (3.3) u,u ′ velocity in frames S
and S ′
Momentum p = p ′ +mv (3.4)
Angular
momentum
Kinetic
energy
a Frames coincide at t =0.
p,p ′
m
particle momentum
in frames S and S ′
particle mass
J = J ′ +mr ′ ×v +v×p ′ t (3.5) J ,J ′ angular momentum
in frames S and S ′
T = T ′ +mu ′ ·v + 1 2 mv2 (3.6)
T,T ′
kinetic energy in
frames S and S ′
S
vt
S ′
r
r ′
m
Lorentz (spacetime) transformations a
Lorentz factor
) −1/2
γ =
(1− v2
c 2 (3.7)
γ
v
c
Lorentz factor
velocity of S ′ in S
speed of light
Time and position
x = γ(x ′ +vt ′ ); x ′ = γ(x−vt) (3.8)
y = y ′ ; y ′ = y (3.9)
z = z ′ ; z ′ = z (3.10)
(
t = γ t ′ + v c 2 x′) ; t ′ = γ
(t− v )
c 2 x (3.11)
x,x ′
t,t ′
x-position in frames
S and S ′ (similarly
for y and z)
time in frames S and
S ′
S S ′ v
x x ′
Differential
four-vector b
dX =(cdt,−dx,−dy,−dz)
(3.12)
X
spacetime four-vector
a For frames S and S ′ coincident at t = 0 in relative motion along x. See page 141 for the
transformations of electromagnetic quantities.
b Covariant components, using the (1,−1,−1,−1) signature.
Velocity transformations a
Velocity
u x = u′ x +v
1+u ′ xv/c 2 ; u′ x = u x −v
1−u x v/c 2 (3.13)
u ′ y
u y =
γ(1+u ′ xv/c 2 ) ; u y
u′ y =
γ(1−u x v/c 2 )
(3.14)
u ′ z
u z =
γ(1+u ′ xv/c 2 ) ; u z
u′ z =
γ(1−u x v/c 2 )
(3.15)
γ
v
c
u i ,u ′ i
Lorentz factor
=[1−(v/c) 2 ] −1/2
velocity of S ′ in S
speed of light
particle velocity
components in
frames S and S ′
S S ′ u
v
x x ′
a For frames S and S ′ coincident at t = 0 in relative motion along x.